Analysis, Manifolds, and Physics, Part 1

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Gulf Professional Publishing, 1982 - Mathematics - 630 pages
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This reference book, which has found wide use as a text, provides an answer to the needs of graduate physical mathematics students and their teachers. The present edition is a thorough revision of the first, including a new chapter entitled ``Connections on Principle Fibre Bundles'' which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, monopoles, instantons, spin structure and spin connections. Many paragraphs have been rewritten, and examples and exercises added to ease the study of several chapters. The index includes over 130 entries.
  

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Contents

Review of Fundamental Notions of Analysis
1
Mappings
2
Berezinian
3
Relations
5
B Algebraic Structures Definitions
6
Groups
7
Rings
8
Algebras
9
Exterior Differential Systems
229
nature
235
Exterior differential equations
236
First fundamental form induced metric
238
Kitting vector fields
239
Sphere 5
240
Curvature of Einstein cylinder
244
Conformal system for Einstein equations
249

Topology
11
Virasoro representation of ifDiff Sl Ghosts BRST
12
Separation
13
Base
14
Covering and compactness
15
Connectedness
16
Continuous mappings
17
Multiple connectedness
19
Associated topologies
20
Topology related to other structures
21
Metric spaces
23
Cauchy sequence completeness
25
Banach spaces
26
normed vector spaces
27
Banach spaces
29
Hilbert spaces
31
Introduction
32
Measures
33
Measure spaces
34
Dirac adjoint
36
Compactness in weak star topology
40
Homotopy groups general properties
42
Homotopy of topological groups
46
Spectrum of closed and selfadjoint linear operators
47
DIFFERENTIAL CALCULUS ON BANACH SPACES
51
E Key Theorems in Linear Functional Analysis
57
Problems and Exercises
64
Holder spaces
70
Noethers theorems II
71
In variance of the equations of motion
79
Calculus of Variations
82
vector field
83
Implicit Function Theorem Inverse Function Theorem
88
DIFFERENT1ABLE MANIFOLDS
91
Subgroups of Lie groups When are they Lie sub groups?
92
CartanKilling form on the Lie algebra S of a Lie group G
93
Differential Equations
94
Direct and semidirect products of Lie groups and their Lie algebra
95
Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields
102
Homogeneous spaces symmetric spaces
103
Examples of homogeneous spaces Stiefel and Grass man n manifolds
108
Abelian representations of nonabelian groups
110
Differentiable Manifolds Finite Dimensional Case
111
Characters
114
Lie algebras of linear groups
115
B Vector Fields Tensor Fields
117
Graded bundles
118
INTEGRATION ON MANIFOLDS
127
Obstruction to the construction of Spin and Pin bun dles StiefelWhitney classes
134
Groups of Transformations
143
Inequivalent spin structures
150
Cohomology of groups
155
Lifting a group action
161
Short exact sequence Weyl Heisenberg group
163
Cohomology of Lie algebras
167
Problems and Exercises
169
Quasilinear firstorder partial differential equation
171
Exterior differential systems contributed by B Kent Harrison
173
The strain tensor
177
Backlund transformations for evolution equations con tributed by N H Ibragimov
181
Poisson manifolds I
184
The 2sphere
190
B Integration
212
Global properties
222
Characteristic system
250
Conformal transformation of nonlinear wave equations
256
Invariants
261
Masses of nomothetic spacetime
262
Invariant geometries on the squashed seven spheres
263
Harmonic maps
274
Composition of maps
281
Riemannian Manifolds Kahlerian Manifolds
285
KaluzaKlein theories
286
Kahler manifolds CalabiYau spaces
294
B Linear Connections
300
BIS CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE
303
Gauge transformations
305
Hopf fibering S3S2
307
Subbundles and reducible bundles
308
Broken symmetry and bundle reduction Higgs mech anism
310
The EulerPoincare characteristic
321
Geodesies on a proper riemannian manifold
327
Equivalent bundles
334
Universal bundles Bundle classification
335
Problems and Exercises
336
Generalized Bianchi identity
340
Geodetic motion equation of geodetic deviation
344
Cocycles on the Lie algebra of a gauge group Anomalies
349
Causal structures conformal spaces Weyl tensor
350
Vbis Connections on a Principal Fibre Bundle
357
operator
363
DISTRIBUTIONS
373
Sobolev embedding theorem
377
Multiplication properties of Sobolev spaces
386
The best possible constant for a Sobolev inequality on R n 3 contributed by H Grosse
389
Characteristic Classes and Invariant Curvature Integrals
390
HardyLittlewoodSobolev inequality contributed by H Grosse
391
Spaces HsSR
393
Spaces HSS and HsSW
396
Completeness of a ball of XV s in JVf_
398
Distribution with laplacian in L2R
399
Nonlinear wave equation in curved spacetime
400
Problems and Exercises
401
Harmonic coordinates in general relativity
405
Leray theory of hyperbolic systems Temporal gauge in general relativity
407
The geometry of gauge fields
408
Einstein equations with sources as a hyperbolic system
413
Wightman distributions and Schwinger functions contributed by C Doering
414
Spin structure spinors spin connections
415
Distributions
423
Bounds on the number of bound states of the Schrod inger operator
425
Sobolev spaces on riemannian manifolds
428
Subject Index
433
B Distributions
435
Errata to Part I
439
Sobolev Spaces and Partial Differential Equations
486
Problems and Exercises
525
Fourier transforms of expjt and expi
531
Elementary solution of the wave equation
538
B Theory of Degree LeraySchauder Theory
556
Morse Theory
567
Cylindrical Measures Wiener Integral
573
Problems and Exercises
589
A metric on the space of paths with fixed end points
596
References
603
Symbols
611
Index
617
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